Smoke particles in the air typically have masses of the order of 10-16 kg. The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. (a) Find the rootmean-square speed of Brownian motion for a particle with a mass of 3.00 * 10-16 kg in air at 300 K.

Video transcript

Hey everyone. This problem states that brownie in motion refers to the random movement of particles as they collide with fluid particles. Okay, we have a dust particle of mass 2.15 times 10 to the negative 17 kg, which is suspended in an air whose temperature is 308 Calvin. Okay. And we are asked to find the root mean square speed of that Brownie in motion. Okay, now let's recall the root mean square speed. V r. M. S is equal to the square root Of three K. T. Over M. K. Where K is a bolton constant And M is the mass? Okay, well, we know the bolts man constant. We know the temperature and we know the mass. So let's go ahead and substitute those values in to find V. RMS. The root mean square speed that we're looking for. We're gonna get the square root of three kinds of Goldsman constant which is 1. times 10 to the negative 23 joules per Calvin eight times 308 Calvin. The temperature that we're given, the unit of Calvin will cancel. This is divided by the mass which we are given as 2.15 times 10 to the -17 kg. Alright, working this out, we're gonna get the square root of 1. seven 512 times 10 to the negative 20. The units we have left are jewels in the numerator recall. We can write jewel as kilogram meters squared per second squared. Okay, when we do that, we have 2.15 times 10 to the negative kg. Okay. And the unit of kilogram will cancel from the numerator and the denominator. We're left with the unit of meter squared per second squared. Ok. Which when we take the square root is going to give us meters per second, Which is a unit for speed that we want. Okay, so we get this is VRMS and we get VRMS is equal to 0.0244. Okay m per second. And when we look at the possible solutions, they are in millimeters per second. Okay, so multiplying by 1000 to convert into millimeters per second, we get 24.4 millimeters per second for our root mean squared speed. Okay. Of the desk protocol at the temperature of 308 Calvin. Okay, so that's going to be answered. D 24.4 millimeters per second. That's it for this one. I hope this video helped see you in the next one.